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RULES OF

EXPONENTS

Learning Objectives

I can…

Multiply and divide powers using the

properties of exponents

Evaluate expressions with zero and

negative exponents

Apply the properties of integer

exponents to generate equivalent

numerical expressions.

Parts When a number, variable, or expression is

raised to a power, the number, variable, or

expression is called the base and the power

is called the exponent.

nb

What is an Exponent?

An exponent means that you multiply the

base by itself that many times.

For example

x4 = x ● x ● x ● x

26 = 2 ● 2 ● 2 ● 2● 2 ● 2 = 64

The Invisible Exponent

When an expression does not have a visible

exponent its exponent is understood to be 1.

1xx

PRODUCT OF POWERS PROPERTY

When multiplying two expressions with the

same base you add their exponents.

For example

mn bbmnb

42 xx 42x 6x

222 21 22 212 32 8

Try it on your own:

mn bbmnb

73.1 hh

33.2 2

1 073 hh

312 33

2 7333

PRODUCT OF POWERS PROPERTY

QUOTIENT OF POWERS PROPERTY

When dividing two expressions with the

same base you subtract their exponents.

For example

m

n

b

b mnb

2

4

x

x 24x 2x

QUOTIENT OF POWERS PROPERTY

Try it on your own:

m

n

b

b mnb

2

6

.3h

h

3

3.4

3

26h 4h

133 23 9

Zero Exponent

When anything, except 0, is raised to the

zero power it is 1.

For example

0a 1( if a ≠ 0)

0x 1( if x ≠ 0)

025 1

Zero Exponent

Try it on your own

0a 1( if a ≠ 0)

0.1 1 h 1( if h ≠ 0)

01 0 0 0.1 2 1

Negative Exponents

If b ≠ 0, then

For example nb nb

1

2x 2

1

x

23 23

1

9

1

Negative Exponents

If b ≠ 0, then

Try it on your own: nb nb

1

3.1 4 h3

1

h

32.1 5 32

1

8

1

Negative Exponents The negative exponent basically flips the

part with the negative exponent to the

other half of the fraction.

2

1

b

1

2b 2b

2

2

x

1

2 2x 22 x

Math Manners

For a problem to be

completely simplified

there should not be any

negative exponents

Power of a Power

When raising a power to a power you multiply

the exponents

For example

mnb )( mnb

42 )(x 42 x 8x22 )2( 222 42 16

Power of a Power

Try it on your own

mnb )( mnb

23 )(.5 h 23 h 6h

22 )3(.6 223 43 81

Note

When using this rule the exponent can not be

brought in the parenthesis if there is addition

or subtraction

222 )2( x 44 2x

You would have to use FOIL in these cases

Power of a Product When a product is raised to a power, each

piece is raised to the power

For example

mab)( mm ba

2)(xy 22 yx2)52(

22 52 2 54 100

Power of a Product

Try it on your own

mab)( mm ba

3)(.7 hk 33 kh2)32(.8 22 32 94 36

Note

This rule is for products only. When using

this rule, the exponent cannot be brought

in the parenthesis if there is addition or

subtraction

2)2( x 22 2x

You would have to use FOIL in these cases

Power of a Quotient

When a quotient is raised to a power, both

the numerator and denominator are raised

to the power

For example

m

b

am

m

b

a

3

y

x3

3

y

x

Power of a Quotient

Try it on your own

m

b

am

m

b

a

2

.9k

h2

2

k

h

2

2

4.10

2

2

2

4

4

16 4

Mixed Practice

9

5

3

6.1

d

d 952 d 42 d4

2

d

54 42.2 ee 548 e 98e

Mixed Practice

54.3 q 54 q 20q

52.4 lp 5552 pl 5532 pl

Mixed Practice

2

42

)(

)(.5

xy

yx22

48

yx

yx 2428 yx 26 yx

9

253 )(.6

x

xx9

28 )(

x

x

9

16

x

x 91 6 x 7x

Mixed Practice

6523246 )()(.7 pnmnm3 01 21 881 2 pnmnm

3 01 281 81 2 pnm 3 02 03 0 pnm

Mixed Practice

94

56

.8da

da9546 da 42 da

4

2

d

a